Speaker: Orit Raz, TAU

Title: Polynomials vanishing on Cartesian products: The Elekes-Szab\'o problem revisited

Abstract:
---------
In the talk I will sketch the proof of the following recent result (joint
work with Micha Sharir and Frank de Zeeuw):

Let $F$ be a constant-degree trivariate real polynomial, and let
$A,B,C\subset\mathbb{R}$ with $|A|=|B|=|C|=n$.

Then either $F$ vanishes on at most $O(n^{11/6})$ points of the Cartesian
product $A\times B\times C$, or $F$ has a local special group-related form.

This improves a result of Elekes and Szab\'o from 2012.

The result provides a unified tool for improving bounds in various Erd{\H o}s-type 
problems in geometry. I will present several applications of this kind.